Sequence(isi m.math sample paper question)

Question

Suppose, $a>0$. Consider a sequence, $a_{n} = n((ea)^{1/n} - (a)^{1/n})$ . Is the sequence convergent? If it is, then what is the limiting value? I have applied Cauchy's 2nd limit theorem but fails. Plz help me

Answer 1

We have

$$a_n=na^{1/n}(e^{1/n}-1) $$ $$=a^{1/n}\frac {e^{1/n}-1}{\frac {1}{n}} $$

when $n\to +\infty $, $1/n \to 0$. using the well-known limit $$\lim_{X\to 0}\frac {e^X-1}{X}=1$$

we find $$\lim_{n\to+\infty}a_n=\lim_{n\to+\infty} a^{1/n}=a^0=1$$